The class of \(\mathbb {G}\) -regular surfaces is different from the class of Euclidean regular surfaces. In [1]}, the authors give an example of \(\mathbb {H}^1\) -regular surfaces, in \(\mathbb {H}^1\)  identified with \(\mathbb {R}^3\) , that are (Euclidean) fractal sets. Conversely, there are continuously differentiable 2-submanifolds in \(\mathbb {R}^3\)  that are not \(\mathbb {H}^1\) -regular surfaces (see [2]} Remark 6.2 and [3]} Corollary 5.11).
